THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR Q-CURVES AND ODA S PERIOD RELATIONS. To Takayuki Oda on his 60th birthday.

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1 THE BIRCH AND SWINNERTON-DYER CONJECTURE FOR Q-CURVES AND ODA S PERIOD RELATIONS HENRI DARMON, VICTOR ROTGER AND YU ZHAO To Takayuki Oda on his 60th birthday. Contents 1. Introduction 1. Background 4.1. The Birch and Swinnerton-Dyer conjecture in low analytic rank 4.. Oda s period relations and ATR points 5 3. The Birch and Swinnerton-Dyer conjecture for Q-curves Review of Q-curves The main result 9 4. Heegner points on Shimura s elliptic curves An explicit Heegner point construction Heegner points and ATR cycles Numerical examples Proof of Proposition References 1 1. Introduction Let E be an elliptic curve over a number field F and let L(E/F, s) denote its Hasse-Weil L-series. It is widely believed that the Shafarevich-Tate group LLI(E/F ) is finite and that L(E/F, s) extends to an entire function of the complex variable s. The order of vanishing of this function at s = 1, denoted by r an (E/F ), is commonly referred to as the analytic rank of E over F, a terminology justified by the Birch and Swinnerton-Dyer conjecture which asserts that (1) rank(e(f ))? = r an (E/F ). The most convincing evidence for the Birch and Swinnerton-Dyer conjecture is the fact that it is proved when F = Q and L(E, s) := L(E/Q, s) has at most a simple zero at s = 1: Theorem 1.1 (Gross-Zagier, Kolyvagin). If r an (E/Q) 1, then (1) holds for E/Q, and LLI(E/Q) is finite. The proof of Theorem 1.1, which is briefly recalled in Section.1, rests on two key ingredients. The first is the modularity of E, in the strong geometric form which asserts that E is a quotient of the Jacobian of a modular curve over Q. The second is the collection of Heegner points on this modular curve, which satisfies the axioms of an Euler system and provides a valuable bridge between the arithmetic of E and the analytic behaviour of its L-series. Both these ingredients are available in greater generality, most notably when F is a totally real field. In this setting, a modular elliptic curve E over F is said to satisfy the Jacquet-Langlands hypothesis (JL) if either [F : Q] is odd, or there is at least one prime of F at which the automorphic form on GL (A F ) attached to E is not in the principal series. Here, A F stands for the ring of adèles of F. The The research of the second author is financially supported by DGICYT Grant MTM C0-01 and the Grup de recerca consolidat de Catalunya 009 SGR 10. 1

2 HENRI DARMON, VICTOR ROTGER AND YU ZHAO meaning of condition (JL), which only fail to hold for certain elliptic curves of square conductor, is described more concretely in Section.1. Most importantly for the proof of (1), the Jacquet-Langlands hypothesis implies that E is the quotient of the Jacobian of a suitable Shimura curve over F. Shimura curves are equipped with a plentiful supply of CM points, which have been parlayed into the proof of the following number field generalisation of Theorem 1.1. Theorem 1. (Zhang). Let E be a modular elliptic curve over a totally real field F satisfying (JL). If r an (E/F ) 1, then LLI(E/F ) is finite and (1) holds for E/F. Denote by N the conductor of E/F. If either [F : Q] is odd or there exists a prime p of F for which ord p (N) = 1, this is Theorem A of [43]. The full result in which E is only assumed to satisfy (JL) follows from the subsequent strengthening of the Gross-Zagier formula proven in [4]. Both results are discussed further in Section.1. In analytic rank zero, the Jacquet-Langlands hypothesis can be dispensed with: Theorem 1.3 (Longo). Let E be a modular elliptic curve over a totally real field F. If L(E/F, 1) 0, then E(F ) and LLI(E/F ) are finite. Longo s proof [4], building on the approach of [], exploits the theory of congruences between modular forms to realise the Galois representation E[p n ] in the Jacobian of a Shimura curve X n whose level may (and indeed does) depend on n. The Euler system of CM points on X n then gives rise to a collection of p n -torsion cohomology classes which is used to bound the p n -Selmer group of E over F independently of n, and thereby obtain the finiteness of E(F ) and LLI(E/F ). The problem of removing the Jacquet-Langlands hypothesis from Theorem 1. or equivalently, of extending Theorem 1.3 to the case where L(E/F, s) has a simple zero at s = 1 is still very much open. To better understand the difficulty which arises, it is instructive to examine the simplest setting where the Jacquet-Langlands hypothesis fails to hold. Assume for the rest of the introduction that F is a real quadratic field, and consider for now the case where E/F is an elliptic curve of conductor 1. Assuming E is modular, the L-series L(E/F, s) is known to have a functional equation relating its values at s and s, and the sign w E { 1, 1} in this functional equation is always equal to 1 in this case. Let M be any quadratic extension of F, let χ M : G F ±1 be its associated Galois character, and denote by E M the twist of E over F by χ M, so that the L-series L(E/M, s) factors as L(E/M, s) = L(E/F, s)l(e/f, χ M, s) = L(E/F, s)l(e M /F, s). Since E has conductor 1, the sign w EM of the twisted L-series is controlled by the local signs attached to the archimedean places 1 and of F, which are equal to χ M ( 1 ) and χ M ( ) respectively. It follows that w EM = 1 if M is either totally real or CM. In particular, the elliptic curve E is always of even analytic rank over such M. Since an Euler system of Heegner points attached to a quadratic CM extension M/F is only expected to be available when E has odd analytic rank over M, this suggests that the mathematical objects so crucial in Kolyvagin s descent method may be unavailable for elliptic curves of conductor 1. A similar expectation can be derived more generally for all elliptic curves which do not satisfy (JL). Indeed, if E/F is an elliptic curve of square conductor N and M is a quadratic extension of F which is unramified at the primes dividing N, then the same analysis as above reveals that { 0 (mod ), if M is CM or totally real; () r an (E/M) 1 (mod ) otherwise. A quadratic extension M of F which is neither CM nor totally real is called an ATR extension of F. An ATR extension of F thus has two real places and one complex place. (The acronym ATR stands for Almost Totally Real, and is used more generally in [8] to designate quadratic extensions of a totally real field having exactly one complex place.)

3 ODA S PERIOD RELATIONS 3 The present article is motivated by the following specific instance of the Birch and Swinnerton-Dyer conjecture which emerges naturally from the discussion above. Conjecture 1.4. Let E be a (modular) elliptic curve over a real quadratic field F of square conductor N for which w E = 1, and let M/F be an ATR extension of F of discriminant prime to N. If L (E M /F, 1) 0, then E M (F ) has rank one and LLI(E M /F ) is finite. Although it seems tantalisingly close to the setting of Theorem 1., Conjecture 1.4 presents a real mystery and appears to lie beyond the reach of known methods. The difficulty is that, in the absence of the Jacquet-Langlands hypothesis, no natural modular method presents itself in general for constructing the point of infinite order on E M (F ) whose existence is predicted by the Birch and Swinnerton Dyer conjecture. One of the original motivations for singling out Conjecture 1.4 for special study lies in the conjectural construction of a so-called Stark-Heegner point P M? E(M) described in [8]. This construction, which is recalled briefly in Section., involves the images under a complex Abel-Jacobi map attached to the Hilbert modular form associated to E/F of certain ATR cycles indexed by ideals of M. The ATR cycles are null-homologous cycles of real dimension one on the corresponding Hilbert modular surface. It is conjectured in [8] that the point P M? is of infinite order precisely when L (E/M, 1) 0, and that P M? is part of a norm-coherent collection of points defined over abelian extensions of M satisfying Euler-System-like properties. However, progress on Conjecture 1.4 through the theory of ATR cycles is thwarted by our inability to provide much theoretical evidence for the conjectures of [8] at present. The first aim of this note is to study Conjecture 1.4 for the class of elliptic curves E/F which are isogenous over F to their Galois conjugate. Following a terminology that was first introduced by Ribet in [3], these elliptic curves are called Q-curves. Their basic properties are reviewed in Section 3. As explained in that section, the case of Q-curves is ultimately made tractable by the existence of a classical elliptic cusp form f (with non-trivial nebentypus character in general) satisfying L(E/F, s) = L(f, s)l( f, s), leading to a modular parametrisation of E by a classical modular curve X 1 (N) for a suitable N 1. The main theorem of Section 3 is a proof of Conjecture 1.4 for Q-curves: Theorem 1.5. Let E/F be a Q-curve of square conductor N, and let M/F be an ATR extension of F of discriminant prime to N. If L (E M /F, 1) 0, then E M (F ) has rank one and LLI(E M /F ) is finite. The key ingredients in the proof of Theorem 1.5 are a strikingly general recent extension of the theorem of Gross-Zagier obtained by Xinyi Yuan, Shouwu Zhang and Wei Zhang [4] covering cusp forms with possibly non-trivial nebentypus characters, and a strengthening of Kolyvagin s descent method to cover abelian variety quotients of J 1 (N), as worked out in the forthcoming book of Ye Tian and Shouwu Zhang [39]. Section 3 explains how Theorem 1.5 follows from these results and the Artin formalism for certain Rankin L-series. The second part of the article focuses on the special case where the Q-curve E is of conductor 1. Such elliptic curves, which were first systematically studied by Shimura [35], are essentially in bijection with newforms f in S (Γ 0 (N), ε N ) with quadratic Fourier coefficients, where N is the discriminant of the real quadratic field F, and ε N : (Z/NZ) ±1 is the corresponding even Dirichlet character. Section 4 describes the explicit construction, for all quadratic ATR extensions M of F, of a canonical point P M E(M) arising from suitable CM divisors on X 1 (N). The trace to E(F ) of P M is shown to vanish, so that P M can also be viewed as an F -rational point on the twisted curve E M. After explaining how the points P M can be computed complex analytically by integrating the elliptic modular form f, we tabulate these points for a few ATR extensions M of small discriminant. One expects that the height of the point P M is related in a simple way to L (E M /F, 1). Finally, Conjecture 4.7 spells out a precise conjectural relationship between the classical Heegner point P M and the Stark-Heegner point P M? arising from ATR cycles on the Hilbert modular variety. This conjecture, which relates certain complex analytic invariants attached to an elliptic modular form f and its Doi-Naganuma lift, can be viewed as an analogue for Abel-Jacobi maps of Oda s period relations which are formulated in [8]. It is therefore a pleasure to dedicate this article to Takayuki

4 4 HENRI DARMON, VICTOR ROTGER AND YU ZHAO Oda whose work on periods of Hilbert modular surfaces was a major source of inspiration for the conjectures of [8]. It is also a pleasure to thank Xavier Guitart, Ariel Pacetti and David Rohrlich for their comments on a previous version of this manuscript.. Background.1. The Birch and Swinnerton-Dyer conjecture in low analytic rank. We begin by recalling in greater detail the main ideas behind the proofs of Theorems 1.1 and 1.. We start with the assumption that F is a number field and write O F for its ring of integers. Let N O F denote the conductor of the elliptic curve E/F. The proofs of Theorems 1.1 and 1. can be broken up into five steps: (i) Modularity: When F = Q, the main results of [41] and [38] (as completed in [1]) imply that there is a normalised newform f of weight on Γ 0 (N) satisfying L(E, s) = L(f, s). In particular, L(E, s) has an analytic continuation to the left of the half plane Re(s) > 3/, and its order of vanishing at s = 1 is therefore well defined. For general F, the modularity of E/F is just the assertion that L(E/F, s) is the L-series attached to an automorphic representation of GL (A F ). Such a property is predicted to hold, as a (very special) case of the Langlands functoriality conjectures. In spite of the powerful ideas introduced into the subject building on Wiles breakthrough, a proof in the general number field setting still seems a long way off. When F is totally real, modularity can be phrased in terms of modular forms much as in the case F = Q. Namely, E/F is modular whenever there is a normalised Hilbert modular eigenform f of parallel weight on the congruence group Γ 0 (N) SL (O F ) satisfying L(E/F, s) = L(f, s). The methods originating from Wiles work seem well suited to yield a proof of modularity of all elliptic curves over totally real fields. (See for example the works of Skinner-Wiles [37], Fujiwara [10], Jarvis-Manoharmayum [19] and the references therein for an overview of the significant progress that has been achieved in this direction.) Currently, the case which offers most difficulties arises when the residual Galois representation at 3 is reducible. (ii) Geometric modularity: Thanks to the geometric construction of Eichler-Shimura and to Faltings proof of the Tate conjecture for abelian varieties over number fields, the modularity of E in the case where F = Q can be recast as the statement that E is a quotient of the jacobian J 0 (N) of the modular curve X 0 (N) over Q, where N = (N), N 1. A non-constant morphism (3) π E : J 0 (N) E of abelian varieties over Q is called a modular parametrisation attached to E. When F is a totally real field and E/F is known to be modular, the modular parametrisation arising from Eichler-Shimura theory admits no counterpart in general. However, such a modular parametrisation can be obtained when the Jacquet-Langlands hypothesis formulated in the introduction holds. More precisely, as it is explained in [43, 3], hypothesis (JL) implies that E is a quotient of the Jacobian of a suitable Shimura curve X attached to an order in a quaternion algebra over F which splits at exactly one archimedean place of F. That is, there is a non-constant map (4) π E : J(X) E of abelian varieties over F generalising (3). The condition that the automorphic form π = π v attached to E be a principal series representation at a place v of F is satisfied precisely when E acquires good reduction over an abelian extension of F v. For v, the meanings of various conditions on the local representations π v in terms of the behaviour of E over F v are summarised in the table below. (5) π v E/F v ord v (N) Unramified principal series Good reduction over F v 0 Principal series Good reduction over an abelian extension of F v even Steinberg Potentially multiplicative reduction over F v 1 or Supercuspidal Otherwise We refer the reader to [1, p. 73], [30], [33, Prop. ], [34, Prop. and 3] for proofs of these statements. (Note that, although in the latter article the ground field is assumed to be F = Q, the

5 ODA S PERIOD RELATIONS 5 results remain valid for arbitrary F as the questions at issue are purely local). See [30] for the behaviour at places v above. In particular, an elliptic curve which fails to satisfy hypothesis (JL) is necessarily of square conductor. The converse is not true, but note that it also follows from the table that all elliptic curves of conductor 1 over a totally real number field of even degree fail to satisfy (JL). We will often restrict our attention to elliptic curves of square conductor, thus encompassing all elliptic curves that do not satisfy hypothesis (JL). (iii) Heegner points and L-series: Suppose first that F = Q and let K be an imaginary quadratic field of discriminant relatively prime to N and satisfying the Heegner hypothesis: O K has an ideal N of norm N satisfying O K /N Z/NZ. An ideal N of K with this property is sometimes called a cyclic ideal of norm N. When the Heegner hypothesis is satisfied, it can be shown that the L-function L(E/K, s) has sign 1 in its functional equation, and therefore vanishes at s = 1. The CM points on X 0 (N) attached to the moduli of elliptic curves with complex multiplication by O K, and their images under π E, can be used to construct a canonical point P K E(K): the so-called Heegner point on E attached to K. The main result of [15] expresses L (E/K, 1) as a multiple by a simple non-zero scalar of the Néron-Tate height of P K. In particular, the point P K is of infinite order if and only if L (E/K, 1) 0. In the setting where F is a totally real field, the Shimura curve X is equipped with an infinite supply of CM points enjoying properties similar to their counterparts on modular curves. The auxiliary field K is now a totally complex quadratic extension of F satisfying a suitable Heegner hypothesis relative to X. The CM points attached to K can be used to construct a canonical point P K E(K) as in the case F = Q. A general extension of the Gross-Zagier theorem ([43, Theorem C]) to this context relates the height of P K to the derivative L (E/K, 1). In particular, the point P K is of infinite order precisely when L(E/K, s) has a simple zero at s = 1. We emphasise that this more general Heegner point construction relies crucially on E/F satisfying hypothesis (JL). (iv) The Euler system argument: The Heegner point P K does not come alone, but can be related to the norms of algebraic points on E defined over abelian extensions of K. Using this fact, it is shown in [0] in the case F = Q that the point P K, when it is of infinite order, necessarily generates E(K) Q. Koyvagin s argument extends without essential difficulties to the context of Shimura curves over totally real fields (cf. [], [43, 7.], or the forthcoming book [39]). (v) Descending from K to F : Assume first that F = Q. If ord s=1 (L(E, s)) 1, the analytic nonvanishing results of [3] or [7] produce an imaginary quadratic field K satisfying the Heegner hypothesis, and for which L(E/K, s) has a simple zero at s = 1. By the Gross-Zagier theorem, the Heegner point P K generates E(K), and its trace therefore generates E(Q). The known properties of the Heegner point P K imply in particular that its trace to Q vanishes when L(E, 1) 0, and is of infinite order when L(E, 1) = 0. Theorem 1.1 for E/Q follows from this. The proof of Theorem 1. is deduced similarly, by noting that the analytic non-vanishing results of [3] or [40] generalize to any number field and again produce a totally complex imaginary quadratic extension K/F satisfying the Heegner hypothesis for which ord s=1 (L(E/K, s)) = 1... Oda s period relations and ATR points.. This section briefly recalls the main construction of [8] which to any ATR extension M of F (satisfying a suitable Heegner condition) associates a point P M E(C) belonging conjecturally to E(M). The points P M arise by considering the images of certain non-algebraic cycles on Hilbert modular varieties under a map which is formally analogous to the Griffiths-Weil Abel-Jacobi maps on null-homologous algebraic cycles. The general setting. We begin by treating a more general context where F is a totally real field of degree r + 1. (This extra generality does not unduly complicate the notations, and may even clarify some of the key features of the construction.) Fix an ordering v 0, v 1,..., v r of the r+1 distinct real embeddings of F. Let E be an elliptic curve over F, and let E j := E vj R (0 j r) be the r+1 elliptic curves over R obtained by taking the base change of E to R via the embedding v j. To ease the exposition, we will make the following inessential assumptions:

6 6 HENRI DARMON, VICTOR ROTGER AND YU ZHAO (1) The field F has narrow class number one; () the conductor of E/F is equal to 1 (i.e., E has everywhere good reduction). (For a more general treatment where these assumptions are significantly relaxed, see for instance the forthcoming PhD thesis [11].) The Hilbert modular form G on SL (O F ) attached to E is a holomorphic function on the product H 0 H 1 H r of r+1 copies of the complex upper half plane, which is of parallel weight (,,..., ) under the action of the Hilbert modular group SL (O F ). The latter group acts discretely on H 0 H r by Möbius transformations via the embedding (v 0,..., v r ) : SL (O F ) SL (R) r+1. Because of this transformation property, the Hilbert modular form G can be interpreted geometrically as a holomorphic differential (r+1)-form on the complex analytic quotient (6) X(C) := SL (O F )\(H 0 H 1 H r ), by setting ω hol G := (πi)r+1 G(τ 0,..., τ r )dτ 0 dτ r. It is important to replace ωg hol by a closed, but non-holomorphic differential (r+1)-form ω G on X(C). When r = 1, the differential ω G is defined by choosing a unit ɛ O F of norm 1 satisfying and setting ɛ 0 := v 0 (ɛ) > 0, ɛ 1 := v 1 (ɛ) < 0, ω G = (πi) (G(τ 0, τ 1 )dτ 0 dτ 1 G(ɛ 0 τ 0, ɛ 1 τ 1 )dτ 0 d τ 1 ). For general r, one defines ω G similarly, but this time summing over the subgroup of O F /(O+ F ) of cardinality r consisting of units ɛ with v 0 (ɛ) > 0. Note that the closed (r+1)-form ω G is holomorphic in τ 0, but only harmonic in the remaining variables τ 1,..., τ r. The justification for working with ω G rather than ωg hol lies in the following statement which is a reformulation of a conjecture of Oda [8] in the special case of modular forms with rational fourier coefficients: Conjecture.1 (Oda). Let { Λ G := γ } ω G, γ H r+1 (X(C), Z). Then Λ G is a lattice in C and the elliptic curve C/Λ G is isogenous to E 0. In [8], this conjecture is shown to hold for Hilbert modular forms which are base change lifts of classical elliptic modular forms, which corresponds to the case where the associated elliptic curve E is a Q-curve. But it should be emphasised that no Q-curve hypothesis on E is necessary in Conjecture.1. Let Z r (X(C)) := Null-homologous cycles of real dimension r on X(C) Conjecture.1 makes it possible to define an Abel-Jacobi map (7) AJ G : Z r (X(C)) E 0 (C), by choosing an isogeny ι : C/Λ G E 0 (C), and setting ( ) (8) AJ G ( ) := ι ω G, (for any with = ). Note that the domain Z r (X(C)) of AJ G has no natural algebraic structure, and that the map AJ G bears no simple relation (beyond an analogy in its definition) with the Griffiths-Weil Abel-Jacobi map on the Hilbert modular variety X. ATR Cycles. Generalising slightly the definitions given in the Introduction to the case r > 1, a quadratic extension M of F is called an ATR extension if. M F,v0 R C, M F,vj R R R, (1 j r).

7 ODA S PERIOD RELATIONS 7 The acronym ATR stands for Almost Totally Real, since an ATR extension of F is as far as possible from being a CM extension, without being totally real. Fix an ATR extension M of F, and let Ψ : M M (F ) be an F -algebra embedding. Then (1) Since M F,v0 R C, the torus Ψ(M ) has a unique fixed point τ 0 H 0. () For each 1 j r, the fact that M F,vj R R R shows that Ψ(M ) has two fixed points τ j and τ j on the boundary of H j. Let Υ j H j be the hyperbolic geodesic joining τ j to τ j. An embedding Ψ : M M (F ) has a conductor, which is defined to be the O F -ideal c Ψ for which Ψ(M) M (O F ) = Ψ(O F + c Ψ O M ). The O F -order O Ψ := O F + c Ψ O M is called the order associated to Ψ. It can be shown that there are finitely many distinct SL (O F )-conjugacy classes of embeddings of M into M (F ) associated to a fixed order O O M, and that the Picard group (in a narrow sense) of O acts simply transitively on the set of such conjugacy classes of embeddings. By the Dirichlet unit theorem, the group is of rank r and preserves the region Γ Ψ := Ψ((O + Ψ ) ) SL (O F ) R Ψ := {τ 0 } Υ 1 Υ r. The ATR cycle associated to the embedding Ψ is defined to be the quotient Ψ := Γ Ψ \R Ψ. It is a closed cycle on X(C) which is topologically isomorphic to an r-dimensional real torus. In many cases, one can show that Ψ is null-homologous, at least after tensoring with Q to avoid the delicate issues arising from the possible presence of torsion in integral homology. (The homological triviality of Ψ always holds, for instance, when r = 1, and follows from the fact that the group cohomology H 1 (SL (O F ), C) is trivial.) Assume from now on that Ψ is homologically trivial, and therefore that it belongs to Z r (X(C)). The following conjecture lends arithmetic meaning to the Abel-Jacobi map AJ G and to the ATR cycles Ψ. Conjecture.. Let Ψ : M M (F ) be an F -algebra embedding of an ATR extension M of F. Then the complex point AJ G ( Ψ ) E 0 (C) is algebraic. More precisely, the isogeny ι in the definition (8) of AJ G can be chosen so that, for all Ψ, AJ G ( Ψ ) belongs to E(H cψ ), where H cψ is the ring class field of M of conductor c Ψ. Furthermore, if Ψ 1,..., Ψ h is a complete system of representatives for the SL (O F )-conjugacy classes of embeddings of M in M (O F ) of a given conductor c, then the Galois group Gal(H c /M) acts (transitively) on the set {AJ G ( Ψ1 ),..., AJ G ( ψh )}. Conjecture. is poorly understood at present. For instance, it is not clear whether the Tate conjecture sheds any light on it. On the positive side, the ATR points that are produced by Conjecture. are genuinely new and go beyond what can be obtained using only CM points on Shimura curves. Indeed, the former are defined over abelian extensions of ATR extensions of totally real fields, while the latter are defined over abelian extensions of CM fields. Most germane to the concerns of this paper, Conjecture. can be used as a basis for the construction of a point P M?? E(M), by letting Ψ 1,..., Ψ h be a complete system of representatives for the SL (O F )- conjugacy classes of embeddings of M in M (O F ) of conductor 1 and setting (9) P? M := AJ G( Ψ1 ) + + AJ G ( Ψh ). 3. The Birch and Swinnerton-Dyer conjecture for Q-curves 3.1. Review of Q-curves. The first goal of the present work is to study Conjecture. for Q-curves, which are defined as follows: Definition 3.1. Let F be a number field and fix an algebraic closure Q of Q containing F. We say that an elliptic curve E/F is a Q-curve if it is isogenous over F to all its Galois conjugates over Q.

8 8 HENRI DARMON, VICTOR ROTGER AND YU ZHAO In the literature, these curves are sometimes referred as Q-curves completely defined over F, reserving the term Q-curve for the wider class of elliptic curves over F which are isogenous over Q to all their Galois conjugates over Q. Q-curves are known to be modular, thanks to the work of Ellenberg and Skinner [ES] (who proved (geometric) modularity of Q-curves under local conditions at 3), now vastly superseded by [3], which implies modularity of all Q-curves as a very particular case. Combining this with the older work of Ribet (cf. [3] for a survey), it follows that Q-curves E/F are arithmetically uniformisable over Q by the classical modular curves X 1 (N). By this, we mean that there exists a non-constant morphism of curves (10) π E : X 1 (N) Q E Q over Q, for some N 1. For simplicity, in this article we shall restrict our attention to Q-curves over a quadratic field F, which represents the simplest non-trivial scenario. However, we believe that the ideas present in this note should allow, with some more effort, to treat more general cases; see the forthcoming Ph.D thesis [45] of the third author. Let F be a quadratic field with ring of integers O F and write Gal (F/Q) = {1, τ}. Let E be a Q-curve over F of conductor N O F. Given a Dirichlet character ε of conductor N, let Γ ε (N) = { ( a b c d ) N c, ε(a) = 1} SL (Z) and let X ε (N) be the modular curve associated to this congruence subgroup. The curve X ε (N) admits a canonical model over Q, and coarsely represents the moduli problem of parametrizing triples (A, C, t) where A is a generalised elliptic curve, C is a cyclic subgroup of order N of A( Q) and t is an orbit in C \ {O} for the action of ker(ε) (Z/NZ). Note that the group (Z/NZ) acts on X ε (N) via the diamond operators, and that the subgroup ker(ε) fixes it. As discussed for example in [14] or [3], the modular parametrisation (10) is particularly wellbehaved when F is quadratic. More precisely, there exists a positive integer N 1, an even Dirichlet character ε : (Z/NZ) {±1} C, and a pair f E, f E S (Γ ε (N)) S (Γ 0 (N), ε) of normalised newforms of weight, level N and nebentypus ε, such that (11) L(E/F, s) = L(f E, s) L(f E, s). In this case, the uniformisation in (10) factors through a modular parametrisation (1) π E : X ε (N) F E F defined over F. Let K f denote the field generated by the Fourier coefficients of f E. It is either Q or a quadratic field. When K f = Q, the elliptic curve E is in fact isogenous to the base change of an elliptic curve defined over Q and question (1) can rather be tackled with the classical techniques reviewed in ; we assume throughout that this is not the case. Hence [K f : Q] = and, letting σ denote the single nontrivial automorphism of K f, we have f E = σ f E. Weil s restriction of scalars A := Res F/Q (E) is an abelian surface of GL -type over Q such that (13) End Q (A) Q K f (and thus is simple over Q), (14) A /F E τ E, and (15) L(A/Q, s) = L(E/F, s) = L(f E, s) L(f E, s). Moreover, for any field extension L/Q, there is a canonical isomorphism (16) A(L) E(F Q L) and in particular A(Q) E(F ). As shown by Carayol, the conductor of A over Q is N, and it follows from [5, Prop. 1] (see also [13, Remark 9] for a more detailed discussion) that the conductor of E/F is N = N 0 O F, where N 0 Z satisfies (17) N = N 0 disc(f ).

9 ODA S PERIOD RELATIONS 9 As we shall now explain, when F is imaginary the problem can be reduced to the classical setting considered by Gross-Zagier and Kolyvagin-Logachev, and presents no mysteries. It is the case of F real that deserves more attention, and to which the main bulk of this note will be devoted. If ε is trivial, then K f is real and F can be either real or imaginary (and indeed both cases occur in examples). As a direct consequence of (11), (16) and the generalization [1] of the work of Kolyvagin to higher dimensional quotients of J 0 (N) over Q, (1) also holds for E/F provided ord s=1 L(f E, s) 1. Assume ε 1 for the rest of this article. Now K f is an imaginary quadratic field. Besides, it follows from an observation of Serre (cf. [3, Proposition 7.]) that F is necessarily real. In fact, F can be computed explicitly from f E as F = Q ker(ε). In particular it follows that ε is the quadratic Dirichlet character associated with F. Let ω N denote the Fricke involution of X ε (N) F defined on the underlying Riemann surface by the rule τ 1 Nτ. It induces an involution on the jacobian J ε(n) F of X ε (N) F which leaves A F stable. We have (18) A F (1 + ω N )A F (1 ω N )A F, where both factors on the right have dimension 1, are isogenous over F and conjugate one each other over Q (cf. [6, 5]). By replacing E by its conjugate if necessary, we shall assume throughout that E = (1 + ω N )A F. It then follows that (1) factors through the following commutative diagram: (19) X ε (N) F J ε (N) F A F π N X ε + (N) J ε + (N) ϕ E E, where we set X + ε (N) := X ε (N) F / ω N and J + ε (N) := (1 + ω N )J ε (N) F. The reader should keep in mind that both are varieties over F, not over Q. 3.. The main result. The goal of this section is to prove Theorem 1.5 of the Introduction. Let M be a quadratic ATR extension of F. Since M has two real places and one complex place, it is not Galois over Q. Let M denote its Galois conjugate over Q, and let M be the Galois closure of M over Q. It is not hard to see that M is the compositum over F of M and M and that Gal(M/Q) is isomorphic to the dihedral group of order 8. The subgroup V F := Gal(M/F ) is isomorphic to a Klein 4-group. The dihedral group of order 8 contains two distinct, non-conjugate subgroups which are isomorphic to the Klein 4-group. This is most easily seen by viewing D 8 as the symmetry group of a square, as in the figure below, in which V F is identified with the subgroup generated by the reflections about the two diagonals. These two reflections can be labeled as τ M and τ M, in such a way that M VF = F, M τm = M, M τ M = M. The second Klein four-group, which shall be denoted V K, is generated by the reflections about the vertical and horizontal axes of symmetry of the square. We label these reflections as τ L and τ L, as shown in the figure below. (0) τ M τ L τ M τ L Now let K := M VK, L := M τl, L = M τ L.

10 10 HENRI DARMON, VICTOR ROTGER AND YU ZHAO These fields fit into the following diagram of field extensions, where each unbroken line indicates an extension of degree : (1) M M M L L F K Let Q χ M, χ M : G F {±1} denote the Galois characters of the real quadratic field F which cut out the extensions M and M, and let χ L, χ L : G K {±1} be the quadratic characters of the imaginary quadratic field K which cut out the extensions L and L. We will often view these characters as idèle class characters defined on A F and on A K respectively. Finally, let ε F and ε K denote the quadratic Dirichlet characters attached to F and K, and let N F Q : A F A Q and NK Q : A K A Q denote the norms on adèles. Proposition 3.. (1) The field K is a quadratic imaginary field. () The characters χ M, χ M, χ L and χ L, viewed as idèle class characters of F and K respectively, satisfy χ M χ M = ε K NQ F ; χ Lχ L = ε F NQ K. (3) The central character of χ M and χ M is ε K, and the central character of χ L and χ L is ε F. (4) The following two-dimensional representations of G Q are isomorphic: Ind Q F χ M = Ind Q F χ M = IndQ K χ L = Ind Q K χ L. Proof. The quadratic field K is of the form Q( d), where d is defined (modulo squares in Q ) by d = N F/Q (α), with M = F ( α). The fact that M is ATR implies that d is a negative rational number, and therefore that K is an imaginary quadratic field. The second part follows directly from the field diagram (1) above. To prove the third part, note that part () implies that the central character of L restricted to the group of norms from K is equal to ε F. (This is because χ L (x) = χ L ( x), where x x is complex conjugation). Class field theory implies that this central character differs from ε F by a power of ε K. But the central character of χ L cannot be ε F ε K since this is an odd Dirichlet character and the central character of a finite order Hecke character of an imaginary quadratic field is necessarily even, because the map from the group of components of R to the group of components of C is trivial. Finally, the proof of part (4) is a simple exercise in representation theory: the four representations that are listed in (4) are all isomorphic to the unique irreducible two-dimensional representation of Gal(M/Q). As at the end of the previous section 3.1, let E be a Q-curve over a real quadratic field F and A = Res F/Q (E), let f = f E S (Γ ε (N)) denote the modular form associated to it and let K f denote the imaginary quadratic field generated over Q by the Fourier coefficients of f. Theorem 3.3 (Tian, Yuan, Zhang and Zhang). Let K be a quadratic imaginary field satisfying the Heegner hypothesis, and let χ : A K C be a finite order Hecke character of K satisfying () χ A Q = ε 1 f. Then (i) The L-function L(f/K, χ, s) vanishes to odd order at s = 1;

11 ODA S PERIOD RELATIONS 11 (ii) If L (f/k, χ, 1) 0, then (A(K ab ) C) χ has rank one over K f Q C, and LLI(A/K ab ) χ is finite. Proof. The modular form f gives rise to a cuspidal automorphic representation π of GL (A Q ) whose central character is ω π = ε f, the nebentypus of f. Condition () ensures that the tensor product of the motives attached to π and χ is self-dual, and therefore the L-function L(f/K, χ, s) = L(π, χ, s 1 ) satisfies a functional equation whose central critical point is s = 1; since the discriminant of K is relatively prime to N, the sign of this functional equation is ( 1) Σ, where Σ = {primes l inert in K such that ord l (N) is odd} { }. The Heegner hypothesis satisfied by K says that Σ = { } and thus the sign is 1; this implies (i). As a consequence of (13), the complex vector space (A(K ab ) C) χ is naturally a K f Q C-module. Part (ii) is a theorem of Tian-Zhang [39] which follows as a corollary of [4, Theorem 1.3.1] by applying Kolyvagin s method. Since [39] is not currently available, the reader may consult [4, Theorem 1.4.1] and, for the precise statement quoted here, [44, Theorem 4.3.1]. We are now ready to prove Theorem 1.5 of the introduction. Theorem 3.4. Let E be a Q-curve over F of square conductor N, and let M/F be an ATR extension of F of discriminant prime to N. If L (E M /F, 1) 0, then E M (F ) has rank one and LLI(E M /F ) is finite. Proof. By (11) and the Artin formalism for L-series, (3) L(E M /F, s) = L(E, χ M, s) = L(f χ M /F, s) = L(f Ind Q F χ M, s). By part 4 of Proposition 3., (4) L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f χ L /K, s). It follows from (3) and (4) that (5) L (E M /F, 1) = L (f χ L /K, 1) = L (f σ χ L /K, 1). Therefore the two rightmost expressions in (5) are non-zero by assumption, so that the product L(f χ L /K, s)l(f σ χ L /K, s) = L(A/K, χ L, s) vanishes to order exactly = [K f : Q] at s = 1. By Theorem 3.3, it follows that A(L) is of rank two, where A(L) denotes the subgroup of the Mordell-Weil group of A(L) of points whose trace to K is trivial. In particular, the Galois representation Ind Q K χ L occurs in A(Q) C with multiplicity. Hence, invoking once again part 4 of Proposition 3., we find that rank(a(m) ) =. But since M contains F and since A is isogenous over F to E E, it follows that rank(e(m) ) = rank(e M (F )) = 1. The result about the ranks follows. The result about the finiteness of LLI(E M /F ) follows in the same way from part (ii) of Theorem Heegner points on Shimura s elliptic curves Implicit in the proof of Theorem 3.4 (via the use that is made of it in the proof of Theorem 3.3) is the construction of a Heegner point P M E M (F ) arising from the image of certain CM divisors on X ε (M) via the modular parametrisation (10). We now wish to make this construction explicit in the case where the Q-curve E has everywhere good reduction over the real quadratic field F. The Q-curves with this property are sometimes called Shimura elliptic curves because they were first systematically considered by Shimura. More precisely, in [35] it is shown how to associate a Shimura elliptic curve over F = Q( N) to any classical elliptic modular form f S (Γ 0 (N), ε N ) with quadratic fourier coefficients. (Cf. also (17).) It will be assumed throughout this chapter that E/F is a Shimura elliptic curve, and that f is the corresponding elliptic modular form. We also assume for simplicity that N is odd, and thus square-free.

12 1 HENRI DARMON, VICTOR ROTGER AND YU ZHAO Remark 4.1. According to calculations performed by the third author using PARI [9] (extending the data gathered in [6, 6], [31] in the range N 1000), there exists an eigenform f S (Γ 0 (N), ε N ) of prime level 1 < N < 5000, with fourier coefficients in a quadratic imaginary extension K f = Q( d) for precisely the values of N and d listed in the following table. N d N d Associated to each such eigenform there is a Shimura elliptic curve over F = Q( N). Furthermore, according to computations due to Cremona, Dembelé, Elkies and Pinch, there are only four primes N in the range [1, 1000] for which there exists an elliptic curve with good reduction everywhere over F = Q( N) which is not a Q-curve, namely, N = 509, 853, 99 and 997. It is hard to predict whether the preponderance of Shimura elliptic curves among elliptic curves of conductor one will persist or is merely an artefact of the relatively low ranges in which numerical data has been gathered. Note that it is not even known whether there exist infinitely many Shimura elliptic curves over real quadratic fields, while it is a theorem of S. Comalada [5] that there are infinitely many elliptic curves over real quadratic fields with good reduction everywhere An explicit Heegner point construction. Let us recall the diagram of field extensions introduced in (1): (6) M M M Q( N, d) L L F Q( Nd) K Q The following lemma is crucial in constructing the point P M E(M) = E M (F ) explicitly. Lemma 4.. Let M be an ATR extension of F and let K be the quadratic imaginary field attached to M as in the diagram (6). Then K has a (canonical) ideal N of K of norm N. In particular, all the prime divisors of N are either split or ramified in K. Proof. The conductor-discriminant formula combined with part 4 of Proposition 3. show that Therefore, after setting disc(f )Nm F/Q (disc(m/f )) = disc(k)nm K/Q (disc(l/k)). N ram = gcd(n, disc(k)), N split = N/N ram, N ram = (N ram, disc(k)), we find that N := N ram N split gives the desired ideal of norm N. Let A K denote the ring of adèles of K, and let N split = (N split, disc(l/k)), Ô K := v O v denote the maximal compact subgroup of the group A K,fin of finite idèles of K. Given a rational integer c 1, (c, N) = 1, we define U c = Ẑ (1 + côk)c A K.

13 ODA S PERIOD RELATIONS 13 By class field theory, the quotient G c := A K /(K U c ) is identified with Gal(H c /K), where H c is the ring class field of K of conductor c. As a piece of notation, we shall write H c for the ring class field attached to the order in K of conductor c 1 and write K a for the ray class field of conductor a. Define U + c = {β U c such that (β) N ker(ε) (Z/NZ) }, U c = {β U c such that (β) N ker(ε) (Z/NZ) }, and Ũc = U c + U c. Here (β) N denotes the image of the local term of the idèle β at N in the quotient O N /(1 + N O N ) (Z/NZ). Similarly for N. This way we can regard the character ε as having source either O N or O. N Set G c := A K /(K Ũ c ) = Gal( H c /K), where H c is a biquadratic extension of the ring class field H c. It can be written as H c = L c L c, where L c (resp. L c) is the class field attached to U + c (resp. U c ). Proposition 4.3. The relative discriminant of L/K factors as d(l/k) = c N, where c is a positive integer such that L L c and L L c and thus M H c. More precisely, we have c = t c 0 where 0 t 3 and c 0 is odd and square-free. The proof of this proposition is an exercise in class field theory, which we relegate to 4.4 for the convenience of the reader. We now explain how to construct a degree zero divisor on X ε (N) defined over H c. To do this, let A c be an elliptic curve satisfying End(A c ) = O c, where O c := Z + co K is the order in K of conductor c. Such a curve, along with its endomorphisms, may be defined over the ring class field H c. The module A c [N ] of N -torsion points is therefore defined over H c, yielding a point P c := [A c, A c [N ]] X 0 (N)(H c ). The action of G Hc := Gal(Q/H c ) on the points of this group scheme gives a Galois representation ρ N : G Hc (Z/NZ). The composition of ρ N with the nebentypus character ε is a quadratic character of G Hc, which cuts out the quadratic extension L c of H c. The point P c lifts to two points P c + and Pc in X ε (N)(L c ) which are interchanged by the action of Gal(L c /H c ); we do not specify the order in which these points are to be taken. Similarly, we can replace the module A c [N ] by A c [N ], mimic the above construction and obtain points P c + and P c defined over L c. Definition 4.4. Let CM(c) = {P + c, P c, P + c, P c } X ε (N)( H c ) be the set of Heegner points on X ε (N) obtained by letting A c run over all isomorphism classes of elliptic curves with CM by O c. If we let h(o c ) denote the cardinality of the group Pic(O c ) of classes of locally free ideals of O c, the cardinality of CM(c) is 4h(O c ). In fact, CM(c) is naturally the disjoint union of the two subsets CM(c) X ε (N)(L c ) and CM(c) X ε (N)(L c), each of cardinality h(o c ). A Heegner point P CM(c) of conductor c may be described by a triple ([a], n, t), where [a] Pic(O c ) is the class of an invertible O c -module of K, n is an integral ideal of O c such that the quotient O c /n is cyclic of order N, t is an orbit for the action of ker(ε) of an element of order N in an 1 /a = Z/NZ. Let C be the quotient of the ray class group of K of conductor cn for which Artin s reciprocity map of global class field theory furnishes a canonical isomorphism rec : C Gal( H c /K).

14 14 HENRI DARMON, VICTOR ROTGER AND YU ZHAO Let O = O c denote the order of conductor c in K. There are natural exact sequences, sitting in the commutative diagram 1 Gal( H c /H c ) Gal( H c /K) res Hc/Hc Gal(H c /K) 1 rec 1 [β 0 ], [β 0 ] C rec rec Pic(O) 1, where the vertical arrows are isomorphisms. Here, β 0 O N and β 0 O N ε(β 0 ) = 1 and ε(β 0 ) = 1. Artin s reciprocity map induces an isomorphism are elements such that Gal( H c /H c ) O N / ker(ε) O / ker(ε) Z/Z Z/Z. N We thus can formally write elements of C as classes of enhanced ideals, which are defined as elements of the form b := β N β N N n, taken up to principal ideals (b) with b K. Here β N and β N belong to K N / ker(ε) and K / ker(ε) respectively, runs over all prime invertible ideals of O not dividing N N, and the exponents n are integers which are almost all zero. We say an enhanced ideal is integral if β N and β N have representatives in O N and O respectively, and n N 0 for all. The image of the class b in Pic(O) is simply the class of the ideal b = N ord N (β N ) N ord N (β N ) N n generated by it. By Shimura s reciprocity law, (7) rec(b)(d) = b 1 D for all b C and all divisors D J ε (N)( H c ) supported on CM(c). On the left hand side we make use of the natural Galois action of Gal( H c /K) on J ε (N)( H c ), via Artin s reciprocity isomorphism. On the right hand side, a class [b] C acts on CM(c) by the rule (8) b P = ([ab 1 ], n, ϕ b (β N t)), where P = ([a], n, t) CM(c), b = β N β N N n is an integral representative of its class and ϕ b : C/a C/ab 1 is the natural projection map. Writing P = [τ] X ε (N)(C) for some τ H, let γ b GL + (Q) be such that b P = [γ bτ]. Besides this action, there is also the diamond involution W ε, acting on P = [τ] X ε (N)(C) as W ε ([τ]) = [γ ɛ τ] and on P = ([a], n, t) CM(c) as (9) W ε (P ) = ([a], n, dt), for γ ɛ = ( a b Nc d ) Γ0 (N) \ Γ ε (N). The cardinality of CM(c) is 4h(O) and it is acted on freely and transitively by the group W N, W ε C M, where we let C M := rec 1 (Gal( H c /M)) C K. Note that the restriction map res Hc/H c induces an isomorphism C M = Pic(O) = Gal(Hc /K). It is our aim now to define a point P M E(M) (and thus also, by conjugation over F, a point P M E(M )) on the elliptic curve E, rational over the ATR extension M/F. We shall construct P M as a suitable linear combination of certain points P L A(L) and P L A(L ) on the abelian surface A = Res F/Q (E). These points are defined as the trace to L of the projection of P c + X ε (N)(L c ) (respectively of P + c X ε (N)(L c)) on A. Before doing so, we first observe that choosing Pc = W ε (P c + ) instead of P c + (and similarly P c ) is unimportant for our construction, as the next lemma shows that both lead to the instead of P c + same point on A up to sign and torsion. Recall the canonical projection π f : J ε (N) A defined over Q and reviewed in (19), which can be composed with the natural embedding of X ε (N) into its jacobian J ε (N) given by the map P P i. By an abuse of notation, we continue to denote by π f this composition. Lemma 4.5. For any P X ε (N)(Q), the point π f (P ) + π f (W ε (P )) belongs to A(F ) tors. Proof. There is a natural decomposition S (Γ ε (N)) = S (Γ 0 (N)) S (Γ 0 (N), ε) corresponding to the eigenspaces of eigenvalue ±1 with respect to the action of the involution W ε. The rule f(z) f(z)dz yields an identification of S (Γ ε (N)) with the space of holomorphic differentials on X ε (N) C. Via this

15 ODA S PERIOD RELATIONS 15 isomorphism, π f H0 (Ω 1 A ) is contained in S (Γ 0 (N), ε). Consequently, π f (P i ) = π f (W ε (P i )) and π f (P ) + π f (W ε (P )) = π f (P i ) + π f (W ε (P ) i ) = π f (P i ) + π f (W ε (P ) W ε (i )) + π f (W ε (i ) i ) = π f (W ε (i ) i ). This last expression is a torsion point on A(F ) by the Manin-Drinfeld theorem which asserts that degree zero cuspidal divisors on a modular curve give rise to torsion elements in its Jacobian. We now set P L = Tr Lc/L(π f (P + c )) A(L). Note that τ M (P + L ) is either equal to Tr L /L (π c f (P c+ )) or to Tr L c /L (π f (P c )). Without loss of generality, assume that τ M (P + L ) = Tr L /L (π c f (P c+ )) and denote it by P L. Set and define u = if K = Q( 1) and c = 1; 3 if K = Q( 3) and c = 1; 1 otherwise, P A,M := 1 u (P L + P L), P A,M := 1 u (P L P L). The construction of the point P A,M is illustrated in the figure below. τ M P A,M τ L P L P A,M τ M P L P L τ L P A,M P L P A,M This figure suggests and it is indeed easy to check that P A,M A(M), P A,M A(M ). Recall that the morphism ϕ F : A F E introduced in (19) is defined over F M, and therefore that the point P M := ϕ F (P A,M ) belongs to E(M). As a by-product of our explicit construction we obtain the following analytic formula for calculating the point P M. Theorem 4.6. Let τ c, τ c H be elements representing the Heegner points P c +, P + c X ε (N)( H c ). Set (30) z M = [ γb τ c ] γb τ c (f E (τ) + f WN E (τ)) dτ + (f E (τ) + f WN E (τ)) dτ. i i b C M Then P M = η(z M ) where η is the Weierstrass parametrization (31) η : C/Λ E E(C), η(z) = ( (z), (z)). Here, is the Weierstrass function associated with the lattice of periods { } Λ E := (f E + f WN E )dτ δ where δ H 1 (X ε (C), Z) runs over the cycles of X ε (C) such that δ (f E f E WN )dτ = 0.

16 16 HENRI DARMON, VICTOR ROTGER AND YU ZHAO 4.. Heegner points and ATR cycles. The main conjecture that will be formulated in this section relates the Heegner point P M with the Stark Heegner point P M? arising from ATR cycles. Recall that Gal(F/Q) = {1, τ} and D F = disc(f ). Let also c E/F (resp. c E τ /F ) denote either the real period or twice the real period of E/R (resp. of E τ /R) depending on whether E(R) (resp. E τ (R)) is connected or not. Conjecture 4.7. The ATR point P M? is of infinite order if and only if P M is of infinite order and L(E/F, 1) 0. More precisely, (3) P? M = s l P M where l Q, which depends only on (E, F ) and not on M, satisfies l L(E/F, 1) =, with Ω E/F = c E/F c Eτ /F Ω E/F D 1/ F E tor(f ), and s Z depends on M. Below we collect some numerical data in support of conjecture 4.7. Besides the numerical evidence, Conjecture 4.7 is also motivated by the equality L (E/M, 1) = L(E/F, 1)L (E/F, χ M, 1) as recalled in the proof of Theorem 3.4, and by the facts that (1) the Néron-Tate height of the Stark-Heegner point P M? is expected (cf. [8]) to be related in a simple way to L (E/M, 1). () the extension of the Gross-Zagier formula proved in [4] should in principle lead to an analogous relationship between the Heegner point P M and the derivative L (f/k, χ L, 1) = L (f/f, χ M, 1) = L (E/F, χ M, 1). It would be interesting to formulate a precise recipe predicting the power of that arises as a fudge factor in (3). The authors have not made a serious attempt to do this Numerical examples. For N = 9, 37, 41 it is known (cf. [36] and [6, 6]) that there is a unique Shimura elliptic curve defined over F = Q( N) up to isogeny over F. The aim of this section is to provide numerical evidence for conjecture 4.7, which we have gathered by explicitly computing the points P M and P M? for several ATR extensions M/F on each of these three elliptic curves. The computation of the Heegner point P M was performed with the software package PARI [9] by exploiting formula (30) and the material in [36] and [6, 6] to produce a complex invariant z M C/Λ E mapping to P M under the Weierstrass uniformisation. Similarly, the ATR point P M? was computed by following the method explained in [8]. In fact, for our experiments it was sufficient to compute the element z M? C/Λ E mapping to P M? under the Weierstrass uniformisation. For several values of M, the invariants z M and z M? were calculated to roughly 50 digits of decimal accuracy, and the constants s and l in (3) could then be obtained by picking a basis (e 1, e ) for Λ E and searching for a linear dependence relation with small integer coefficients between the four complex numbers z M, z M?, e 1 and e, using Pari s lindep command. This approach represents a dramatic improvement over the one that had to be followed in [8], in which only the point P M? was computed. In practice, recognizing P M? as an algebraic point using standard rational recognition programs is difficult once the height of P M? becomes large. The authors of [8] were forced instead to perform an independent search for a generator of E(M) a computationally difficult and time-consuming task in order to check that P M? indeed agreed with a point of small height on E(M) to within the calculated decimal accuracy. The new approach based on the Heegner point P M makes the experimental verifications of [8] much more systematic and efficient, and allows them to be carried out for much further ranges. In the tables below, we have followed almost the same notations as in [8, 3]. In particular, we have written M = F (β) with β F, and denoted by D M = Nm F/Q (disc(m/f )) the absolute discriminant. (Note that K is used in [8] to denote the field that we call M in the present work.) As before, L/K denotes the quadratic extension sitting in the Galois closure of M as in the field diagram (6), and we denote D K = disc(k/q) and D L = Nm K/Q (disc(l/k)) and l Q is the value we found numerically